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those volumes into a continued series. To each number also will be found annexed, in the Indian figures inclosed in a parenthesis, the number of the same proposition as it was numbered in the first miscellaneous treatise of the year 1787. The additional theorems are as follow.
OF THE ELLIPSE.
THEOREM XII (5). The Difference between the Semi-transverse and a Line
drawn from the Focus to any Point in the Curve, is equal to a Fourth Proportional to the Semi-transverse, the Distance from the Centre to the Focus, and the Distance from the Centre to the Ordinate belonging to that Point of the Curve.
That is, AC - FE = CI, or FE = AI; and fe – AC = ci, or fe = BI. Where CA :cf :: CD: CI the 4th proportional to ca, CF, CD.
For, draw AG parallel and equal to ca the semi-conjugate; and join CG meeting the ordinate de in H. Then, by theor. 2, CA2 : AG? :: CA’ – CD’ : DE?: and, by sim. tri. CA? : AG:: CA? CD2 : AGZ DH; consequently DE? = AG? DH = ca’ – DH?. Also FD = cfi CD, and FD2
2cf. CD + CD2; but by right-angled triangles, PD? + DE = Fe; therefore FE? = CF + ca? 26F.CD + CD DH”. But by theor. 4, ca + CF CA and, by supposition, 2CF. CD = 2CA, CI; theref. Fe = CA 2CA, CI + CDC DH. But by supposition, CA’: CD2 :: cf? or CA – AG? : CI°; and, by sim. tri. CA? ; CD2 :: CA2 AG? : CD
Corol. 1. Hence ci or CA
CA, CF, CD.
Corol. 2. And fe FE = 2C1; that is, the difference between two lines drawn from the foci, to any point in the curve, is double the 4th proportional to CA, CF, CD.
THEOREM XIII (11).
If a Line be drawn from either Focus, Perpendicular to a Tangent to any Point of the Curve; the Distance of their Intersection from the Centre will be equal to the Semitransverse Axis.
That is, if FP, fp be perpendicular to the tangent TPP, then shall CP and cp be each equal T
to CA or CB.
FE is a 4th proportional to
For, through the point of contact E draw FE, and ƒɛ meeting FP produced in G. Then, the GEP FEP, being each equal to the fep, and the angles at P being right, and the side PE being common, the two triangles GEP, FEP are equal in all respects, and so GE FE, and GP = FP. Therefore, since FP=FG, and FCF, and the angle at AB, that is CP=
F common, the side cp will be
CA or CB. Q.Ë.D. Corol. I. A circle described on the transverse axis, as a diameter, will pass through the points P, P; because all the lines CA, CP, cp, CB, being equal, will be radii of the circle. Corol. 2. CP is parallel to fE, and cp parallel to FE.
Corol. 3. If at the intersections of any tangent, with the circumscribed circle, perpendiculars to the tangent be drawn, they will meet the transverse axis in the two foci. That is, the perpendiculars PF, pf give the foci F, f.
THEOREM XIV (12).
The equal Ordinates, or the Ordinates at equal Distances from the Centre, on the opposite Sides and Ends of an Ellipse, have their Extremities connected by one Right Line passing through the Centre, and that Line is bisected by the Centre.
That is, if coce, or the crdinate DE = G# ;
For when CD = CG, then also is DE GH by cor. 2, th. 1. But the 4D 4G, being both right angles; therefore the third side CE = CH, and the 4 DCE = 4&CH, and consequently ECH is a right line.
Corol. 1. And, conversely, if ECH be a right line passing through the centre; then shall it be bisected by the centre, or have CE = CH; also DE will be ≈ GH, and CD = CG.
Corol. 2. Hence also, if two tangents be drawn to the two ends E, и of any diameter EH; they will be parallel to each other, and will cut the axis at equal angles, and at equal distances from the centre. For, the two CD, CA being equal to the two CG, CB, the third proportionals CT, cs will be equalalso; then the two sides CE, CT being equal to the two CH, cs, and the included angle ECT equal to the included angle HCS, all the other corresponding parts are equal: and so the T=48, and тE parallel to HS.
Corol. 3. And hence the four tangents, at the four extremities of any two conjugate diameters form a parallelogram circumscribing the ellipse, and the pairs of opposite sides are each equal to the corresponding parallel conjugate diameters. For, if the diameter eh be drawn parallel to the tangent TE or HS, it will be the conjugate to EH by the definition; and the tangents to e, h will be parallel to each other, and to the diameter EH for the same reason.
THEOREM XV (13).
If two Ordinates ED, ed be drawn from the Extremities E, e, of two Conjugate Diameters, and Tangents be drawn to the same Extremities, and meeting the Axis produced in T and R;
Then shall CD be a mean proportional between cd, dr, and cơ a mean proportional between CD, DT.
For, by theor. 7,
Corol. 3. Also cd2
Corol. 1. Hence CD: cd :: CR: CT.
Corol. 2. Hence also CD : cd :: de : DE. And the rectangle CD. DE
That is, DA and da For, by theor. 7, and by div.
and by comp.
DT: cd: CT
In like manner,
CA :: CA
cd :: cd : DT.
Or cd a mean proportional between CD, DT;
cd. de, or ▲ CDE = A cde.
THEOREM XVI (14).
The same Figure being constructed as in the last Theorem, each Ordinate will divide the Axis, and the Semi-axis added to the external Part, in the same Ratio.
[See the last fig.]
DT:: DC: db,
CD: CA :: CA: CT,
Q. E. D
Corol. 1. Hence, and from cor. 3 to the last, it is,
Q. E. D.
Ad. dB = CA2 ·
Corol. 2. Hence also, CA2= CD2 + cď2, and ca2 DE2 + de2.
Corol. 3. Further, because CA2: ca2:: AD.DBor cď2: DE2, therefore CA: ca :: cd: DE. likewise CA: ca :: CD : de.
THEOREM XVIi (15).
If from any Point in the Curve there be drawn an Ordinate, and a Perpendicular to the Curve, or to the Tangent at that Point: Then, the
Dist. on the Trans, between the Centre and Ordinate, CD:
CA: ca2 :: DC: DP.
For, by theor. 2, CA2 : ca2 :: AD.DB: DE,
But, by rt. angled As, the rect. TD..DP = DE2;
That is, the triangle CET the triangle CAN.
THEOREM XVIII (18).
If there be Two Tangents drawn, the One to the Extremity of the Transverse, and the other to the Extremity of any other Diameter, each meeting the other's Diameter produced; the two Tangential Triangles so formed, will be equal.
For, draw the ordinate DE. Then
CA :: CE : CN;
CA: CA: CT;
: CT:: CE CN.